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Questions about the properties of functions of the form $\sum_{n=0}^{\infty}a_n (x-c)^n$, where the $a_n$ are real or complex numbers, and $x$ is real or complex.

A series of the form $$\sum_{n=0}^{\infty} a_n (x-c)^n$$ is called a power series, and can be used to expand functions. The center $c$ is often $0$ and the radius of convergence $R$ is given by $R = \left(\limsup\limits_{n\to\infty}\sqrt[n]{|a_n|}\right)^{-1}$.

Power series for some common functions are: \begin{align} \frac{1}{1-x}&=\sum_{n=0}^\infty x^n\quad(|x|\lt1)\\\ \ln(1+x)&=\sum_{n=0}^\infty\frac{(-1)^nx^{n+1}}{n+1}\quad(|x|\leq 1, x\neq -1)\\\ \arctan(x)&=\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{2n+1}\quad(|x|\leq 1,x\neq \pm i)\\\ \tan(x)&=\sum_{n=1}^{\infty}\frac{|B_{2n}|(4^n-1)4^n }{(2n)!}x^{2n-1}\quad(|x|< \pi/2)\\\ \sin(x)&=\sum_{n=0}^\infty\frac{(-1)^n x^{2n+1}}{(2n+1)!}\\\ \cos(x)&=\sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!}\\\ e^x &= \sum_{n=0}^\infty\frac{x^n}{n!}\\\ \end{align}

If convergence is not an issue or if you are working over a different domain than $\mathbb{R}$ or $\mathbb{C}$, consider using the tag instead.